nLab 0-sphere

Idea

The 0-sphere $S^0$ is the n-sphere for $n = 0$: The disjoint union of two points,

$$S^0 \;\;\simeq\;\; \ast \sqcup \ast \,.$$

This definition (even if possibly not be familiar from standard texts of point-set topology) is certainly justified by the fact that the 2-element set is clearly:

• the unit sphere inside the real line $\mathbb{R}^1$,

  and as such part of the general pattern that the n-sphere is the unit sphere inside $\mathbb{R}^{n+1}$;

• the topological space/homotopy type whose suspension is equivalently the 1-sphere (the circle):

  $$S^1 \,\simeq\, \mathrm{S} S^0 \,,$$

  thus being the base case of the inductive definition of the $n$-spheres:

  $$S^{n+1} \;\simeq\; \mathrm{S} S^{n} \;\simeq\; \mathrm{S} \mathrm{S} S^{n-1} \;\simeq\; \mathrm{S}^{n+1} S^0 \,.$$

In fact, with suspension $\mathrm{S}X$ understood as the homotopy pushout of the terminal map $X \to \ast$ along itself, one has also

$$S^0 \;\coloneqq\; \mathrm{S} S^{-1}$$

for $S^{-1} \,\coloneqq\, \varnothing$ the empty set.

These spheres of degenerate dimensions play an imprtant role in homotopy theory, see for instance the generating cofibrations in the classical model structure on topological spaces (here).

Of course, the underlying object of $S^0$, simple as it is, plays various other roles, too:

• in topos theory, the two element set plays the role of the subobject classifier among Sets;

• in formal logic this is also known as the classical boolean domain: the set of classical truth values;

  and in data-type theory one also speaks of the type $Bit$ of bits;

• that same type regarded in homotopy type theory again plays the role of the $0$-sphere in the sense of sphere types – see also at boolean domain – In HoTT;

• in category theory the disjoint union of two copies of the terminal object (be it the 2-element set or the corresponding discrete category, etc.) is often denoted “$\mathbf{2}$”

  (cf. e.g. at Stone duality).

Related concepts

• 1-sphere

• 2-sphere

• 4-sphere

• 7-sphere